Transcript Simple Harmonic Motion

AP Physics B: Ch.10 - Elasticity
and Simple Harmonic Motion
Reading Assignment
Cutnell and Johnson,
Physics
Chapter 10
Simple Harmonic motion/systems
Mechanical systems use springs
extensively
Imagine a mechanical car engine – lots
of moving parts – back and forth
movement.
• Prototype for all vibrating systems is
simplest possible case called Simple
Harmonic Motion (SHM)
Simple Harmonic motion
• Important for understanding many
disparate phenomena, e.g., vibration of
mechanical structures (bridges, cars,
buildings)
• Electrical radio receivers
Springs
© John Wiley and Sons, 2004
Obey Hooke’s law in
both extension and
compression, i.e.,
F=kx
where k is spring
constant [N/m]
k is measure of the
stiffness of the spring.
In other words the restoring force of the spring depends
on its strength and how far it is extended
Springs
Newton’s Third Law =>
•
• The restoring force of an ideal spring is
•
F = -k x
• k is spring constant,
• x is displacement from unstrained
length.
• If I exert a force to stretch a spring, the
spring must exert an equal but opposite
force on me: “restoring force”.
Restoring force is always in opposite
direction to displacement of the spring.
Springs
• Stretching an elastic material or spring,
and then releasing leads to oscillatory
motion. In absence of air resistance or
friction we get “Simple Harmonic
Motion”
© John Wiley and Sons, 2004
Springs
The motion of the spring in the diagram
below is said to be simple harmonic
motion.
If a pen is attached to an object in
simple harmonic motion then the output
will be a sine wave.
© John Wiley and Sons, 2004
Springs
Simple harmonic motion is where the
acceleration of the moving object is
proportional to its displacement from its
equilibrium position.
It is the oscillation of an object about its
equilibrium position
The motion is periodic and can be described as
that of a sine function (or equivalently a cosine
function), with constant amplitude. It is
characterised by its amplitude, its period and its
phase.
Springs
The motion is periodic and can be
described as that of a sine function (or
equivalently a cosine function), with
constant amplitude. It is characterised by
its amplitude, its period and its phase.
© John Wiley and Sons, 2004
Springs
• Simple harmonic motion equation –
k
ax  
x   2 x
m
•  - is a constant (angular velocity)
© John Wiley and Sons, 2004
Simple Pendulum
Pivot
•Restoring force acts to pull bob back towards vertical.
•mg sin  is the restoring force

•F= mas so restoring force = mas = mg sin 
L
g
as  
s
Tension
L
l
s
mg cos
mg sin
mg
Arclength 

 

Radius 

Result:
g
as  
s
L
• Compare with acceleration of mass on a
spring.
k
2
ax  
m
x  
x
• Time for one oscillation (T)
g
 pendulum
L
2
L
T
 2

g
Equations are of same form, and we can say for
pendulum is equivalent to simple harmonic motion.
Rigid
beam
Elasticity Experiment
Results
Metal
wire
Vernier
scale
m
mg
mass (kg) Force (N)
extension (mm)
0
0
0
1
9.8
0.14
2
19.6
0.28
3
29.4
0.42
4
39.2
0.56
5
49
0.7
6
58.8
0.9
7
68.6
1.15
8
78.4
1.45
9
88.2
2
10
98
2.7
Elasticity Experiment - Results
Elasticity
extension (mm)
3
2.5
2
1.5
1
0.5
0
0
20
40
60
Force (N)
80
100
•
•
•
•
•
•
•
Definitions
Tensile stress =applied force per unit area
= F/A [N/m2]
Tensile strain = extension per unit length
= L/ Lo
Experiments show that, up to the elastic limit,
Tensile stress  Tensile strain (Hooke’s Law)
i.e.,
F
L
 constant x
A
L0
FL0
 constant 
 "Young ' s Modulus (Y )"
AL
Definitions
• Young’s Modulus = Stress/Stain
FL0

 "Young' s Modulus (Y )"
AL
To measure Young’s Modulus for brass
L0
F
L
Y
 L 
F
A
L0
YA
Measure:
L0 = 1.00 m
A = 0.78 x 10-6 m2
slope = 1.43 x 10-5 m/N
extension (m x 10^-3)
=> Y =
0.8
0.6
0.4
0.2
0
0
10
20
30
F (N)
40
50
Typical Young’s modulus values
Material
Young's modulus (N/m2)
Brass
Aluminium
Steel
Pyrex Glass
Teflon
Bone
9.0 x 1010
6.9 x 1010
2.0 x 1011
6.2 x 1010
3.7 x 108
9.4 x 109 (compression)
1.6 x 1010 (tension)
Damped harmonic Motion
In reality, non-conservative forces always
dissipate energy in the oscillating system.
This is called “damping”.
Lightly damped oscillation
6
x (cm)
4
2
0
-2
-4
-6
0
20
40
60
time (s)
80
100
Heavily damped oscillation
6
x (cm)
4
2
0
-2
-4
0
20
40
60
time (s)
80
100
Over-damped motion
6
x (cm)
5
4
3
2
1
0
0
20
40
60
time (s)
80
100
Resonance
Resonance is the condition where a
driving force can transmit large amounts
of energy to an oscillating object,
leading to a large amplitude motion. In
the absence of damping, resonance
occurs when the frequency of the driving
force matches the natural frequency of
oscillation of the object.
Pushing a swing at the correct
time…/incorrect time…resonance
Summary
• Materials subject to elastic deformation
obey Hooke’s Law
F  kx
• Stretching an elastic material or spring,
and then releasing leads to oscillatory
motion. In absence of air resistance or
friction we get “Simple Harmonic Motion”